On the Capacity Requirement for Arbitrary End-to-End Deadline and Reliability Guarantees in Multi-hop Networks
OOn the Capacity Requirement for ArbitraryEnd-to-End Deadline and Reliability Guaranteesin Multi-hop Networks
Han Deng
Department of ECETexas A&M UniversityCollege Station, TX 77840, USAEmail: [email protected]
I-Hong Hou
Department of ECETexas A&M UniversityCollege Station, TX 77840, USAEmail: [email protected]
Abstract —It has been shown that it is impossible toachieve both stringent end-to-end deadline and reliabilityguarantees in a large network without having completeinformation of all future packet arrivals. In order to maintaindesirable performance in the presence of uncertainty of fu-ture packet arrivals, common practice is to add redundancyby increasing link capacities. This paper studies the amountof capacity needed to provide stringent performance guar-antees. We propose a low-complexity online algorithm andprove that it only requires a small amount of redundancy toguarantee both end-to-end deadline and reliability. Further,we show that in large networks with very high reliabilityrequirements, the redundancy needed by our policy is atmost twice as large as a theoretical lower bound. Also,for practical implementation, we propose a fully distributedprotocol based on the previous centralized policy. Withoutadding redundancy, we further propose a low-complexityorder-optimal online policy for the network. Simulationresults also show that our policy achieves much betterperformance than other state-of-the-art policies.
I. I
NTRODUCTION
Many emerging safety-critical applications, such as In-ternet of Things (IoT) and Cyber-Physical Systems (CPS),require communication protocols that support strict end-to-end delay and reliability guarantees for all packets. Ina typical scenario, when sensors detect unusual eventsthat can cause system instability, they send out this infor-mation to actuators or control centers. This informationneeds to be delivered within a strict deadline for actuatorsor control centers to resolve the unusual events. Thesystem can suffer from a critical fault when a smallportion of packets fail to be delivered on time.Despite the huge literature on quality of service (QoS),there is little work that can provide end-to-end delayand reliability guarantees simultaneously, especially whenpacket arrivals are time-varying and unpredictable. Thelack of progress is mainly caused by two fundamentalchallenges. On one hand, it is obvious that one cannotdesign the optimal network policies without obtainingcomplete knowledge of future packet arrivals and incur-ring high computation complexity. Therefore, practical so- lutions need to rely on online suboptimal policies. On theother hand, in a multi-hop network, the scheduling deci-sion of one communication link will impact the decisionsof subsequent links. The negative effects of suboptimaldecisions by online policies therefore get accumulatedalong the path of multi-hop transmissions. In fact, a recentwork by Mao, Koksal, and Shroff [1] has proved thatthe performance of any online policies deteriorates asthe length of the longest path in the network increases.As a result, no online policy can provide meaningfulperformance guarantees when the size of the network islarge.In order to maintain desirable performance using onlinesuboptimal policies, current practice is to add redundancyinto the system. During system deployment, the capacitiesof communication links are chosen to be larger thannecessary. Such redundancy alleviates the negative im-pacts of suboptimal decisions by online policies. Using thisapproach, a critical question is to determine the amount ofredundancy needed to provide the desirable performanceguarantees. This paper aims to answer this question.We first show that the problem of maximizing thenumber of timely packet deliveries can be formulatedas a linear programming problem when one knows thecomplete knowledge of all future packet arrivals. In thesetting of online policies, some of the parameters ofthis linear programming problem will only be revealedwhen the corresponding packets arrive. Therefore, onlinepolicies need to make routing and scheduling decisionsfor packets without knowing all parameters. On theother hand, we also observe that adding redundancyby increasing link capacities is equivalent to relaxing asubset of constraints in the linear programming problem.Based on these observations, we define a competitiveratio that, given the amount of redundancy, quantifies therelative performance of online policies in comparison tothe optimal offline solution.Using the primal-dual method, we propose an onlinepolicy that achieves good performance in terms of com- a r X i v : . [ c s . PF ] A p r etitive ratio. This policy has several important features:First, when there is no redundancy added to the system,the performance of our online policy is asymptoticallybetter than that of the recent work [1] when the sizeof the network increases. Second, we also show that onlya small amount of redundancy is needed to achieve strictperformance guarantees. Specifically, in order to guaran-tee the timely delivery of at least − θ as many packets asthe optimal solution in a network whose longest path haslength L , our policy only needs to increase link capacitiesby ln L + ln θ times. Finally, we also show that our policycan be implemented with very low complexity.Next, we establish a theoretical lower bound of com-petitive ratio for all online policies. We show that, inorder to guarantee a certain degree of performance, theredundancy needed by our policy is only a small amountaway from the theoretical limit. In particular, when both L and θ , as defined in the previous paragraph, go to infinity,the redundancy needed by our policy is at most twice aslarge as the theoretical limit.We also study online policies when one cannot increasenetwork capacity by adding redundancy. We propose an-other online policy and prove that it is order optimalwith fixed link capacity. Specifically, we show that thisonline policy guarantees to deliver at least O (log L ) asmany packets before their deadlines as the optimal offlinesolution, where L is the maximum route length. As theprevious study [1] has proved no online policy can delivermore than O (log L ) packets without redundancy, our policyis order-optimal.While neither of our online policies need any infor-mation about future packet arrivals to make routing andscheduling decisions, they are centralized algorithms thatrequire tight coordination. For large networks withouta centralized coordinator, we also propose a fully dis-tributed protocol that is inspired by the design princi-ples of our centralized online policies. This distributedprotocol only requires each node to broadcast its localcongestion information very infrequently, and therefore itonly incurs a small amount of communication overhead.When a packet arrives at a source node, the source nodedetermines a suggested route for the packet using itsreceived congestion information, and each link on theroute makes scheduling decisions solely based on its localinformation.All three of our policies are evaluated by simulations.We compare our policies with the widely used earliestdeadline first policy (EDF) and recent policy studied in[1]. Simulation results show that all our policies performbetter than the other two policies. This result is in par-ticular surprising because our distributed protocol evenachieves better performance than the online policy in [1],which is a centralized one.The rest of the paper is organized as follows. SectionII reviews some existing works. Section III introduces oursystem model defines the competitive ratio. Section IV proposes our online policy and studies its competitiveratio and computation complexity. Section V establishesa theoretical lower bound of competitive ratio. SectionVI proposes an order-optimal policy and studies its com-petitive ratio. Section VII proposes a distributed protocolbased on the intuitions of our centralized online policy.Section VIII provides simulation on our proposed algo-rithms and compare them with two other online policies.Finally, Section IX concludes this paper.II. R ELATED W ORK
Online scheduling problem in real-time environmenthas been studied in many previous works. Studies showthat earliest deadline first algorithm (EDF) [2], [3] andleast laxity first algorithm (LLF) [3] achieve the sameperformance as the optimal offline algorithm when thesystem is under-loaded, that is, the optimal offline algo-rithm can serve all jobs in the system. In under-loadedsystem, all jobs enter the system can be served by EDFand we do not need to drop any job when it arrivesat the system. However, in over-loaded system, evenwith optimal offline algorithm, there are still some jobsthat cannot be served. EDF and LLF achieve the sameperformance as the optimal online policy when the systemis over-loaded. Also [3] proved that no online algorithmcan guarantee to serve more than / of the jobs thatcan be served by optimal offline algorithm and providedan algorithm in a uniprocessor system which achieves / service bound. [4], [5] consider admission controlin online scheduling. In [4], when all jobs have equallength, the competitive ratio of deterministic algorithm isbounded by 2. [6] considers the similar model as [4]. Itintroduces a parameter k to indicate the willingness ofa job to have a delay before being served. It shows thatwhen all jobs have equal length, the competitive ratio ofdeterministic algorithm is (1 + 1 / ( (cid:98) k (cid:99) + 1)) -competitiveinstead.In addition, online scheduling with multiple-server casehas also been studied. [7] studies the scheduling of equallength jobs on two identical machines. [8]–[10] studiesthe case with parallel machines. The scheduler need todecide whether to accept or reject a packet and whichmachines is chosen to serve the job. [9] has proposedan algorithm with immediate decision which approaches ee − -competitive when the number of machines is greateror equal to 3. It also provide another lower bound thatdeterministic online algorithm with immediate decisionis no better than 1.8-competitive when there are 2 ma-chines. Later [10] has shown that online algorithm whichmakes immediate decision upon job releasing is boundedby ee − -competitive for multiple machine case.There are also many works studying the schedulingproblem in multihop network. An early study [11] focuseson the problem of packet scheduling with arbitrary end-to-end delay, fix route, and known packet injection rate. Itpropose a distributed algorithm which achieve a certainelay bound. [12] studies the scheduling problem on atree network. Packets arrive at an arbitrary node and theyneed to be transmitted to root node before the deadlines.Any packet that cannot arrive root node within deadline isconsidered lost. Thus this is also a fix route problem. Thegoal is to minimize the total lost packets. Shortest timeto extinction (STE) algorithm is proposed and it is shownto achieve the performance of optimal offline policy. Alsothere are many works studying the end-to-end delay inmultiohop network. Rodoplu et al. [13] have studied theproblem of dynamic estimating end-to-end delay overmulti-hop mobile wireless networks. Sanada Komuro andSekiya [14] have used Markov-chain model to study thestring-topology multi-hop network and analyse the end-to-end throughput and delay. Jiao et al. [15] have studiedthe problem of estimating the end-to-end delay distribu-tion for general traffic arrival model and Nakagami-mchannel model by analyzing packet delay at each hop.Li et al. [16] ] have proposed using expected end-to-end delay for selecting path in wireless mesh networks.The expected end-to-end delay takes both queuing delayand delay caused by unsuccessful wireless transmissions.However, their work only aims at minimizing the averageend-to-end delays, and cannot provide guarantees on per-packet delays.Li and Eryilmaz [17] has studied the end-to-enddeadline constrained traffic scheduling in multihop net-work. They develop algorithms to meet the deadline andthroughput requirement in a wired network. However,they only consider the fix route model and they do notprovide any performance guarantee. Wang et al. [18] havestudied the problem of routing and scheduling on multi-hop wireless sensor network in order to optimize the sys-tem with the constraint of end-to-end delay and proposeda sub-optimal algorithm. Hou [19] proposed a throughputoptimal policy for up-link tree networks with end-to-enddelay constraints and delivery ratio requierement. Thepackets deadlines are the end of the frames in whichthey are generated. Singh and Kumar [20] have proposeda scheduling policy which maximize the throughput formulti-hop wireless networks. However, the paper uses afix-route model and does not consider end-to-end delay.Mao, Koksal and Shroff [1] also considers a fix routeproblem. The network has arbitrary packet arrival andpacket weight. The paper aims to maximize the total cu-mulative weight of packets that reach destination beforetheir deadline. The paper has proved that the competitiveratio of any online policy is no better than O (log L ) , where L is the length of the maximum route. It has also proposedan admission control and packet scheduling policy andshown that it is O ( L log L ) -competitive. Liu and Yang[21] have studied the multi-hop routing problem withhard end-to-end delay and the throughput region. Theyalso assume that packets are required to be deliveredto destination within one frame and the performance isevaluated by simulation. Our work will focus on online routing and scheduling on multi-hop network with end-to-end delay constraint and aim to guarantee both packetdeadline and network delivery ratio.III. S YSTEM M ODEL
We consider a network with multihop transmissions.The network is represented by a directed graph whereeach node represents a router and an edge from one nodeto another represents a link between the correspondingrouters. Packets arrive at their respective source nodesfollowing some unknown sequence. We use M to denotethe set of all packets and L the set of all links. Whena packet m ∈ M arrives at its source node, it specifiesits destination and a deadline. The packet requests to bedelivered to its destination before its specified deadline.Packets that are not delivered on time do not have anyvalue, and can be dropped from the network. We aim todeliver as many packets on time as possible.We assume that time is slotted and numbered by t = { , , , . . . } . Different links in the network mayhave different link capacities, and we denote by C l thenumber of packets that link l can transmit in a time slot.At the beginning of each time slot, each node decideswhich packets to transmit over its links, subject to capacityconstraints of the links. Packets transmitted toward a nodein time slot t will be received by that node at the end ofthe time slot, so that the node can transmit these packetsto subsequent nodes starting from time slot t + 1 .Delivering a packet to its destination before its deadlinerequire determining two things: the route used to forwardthe packet from its source to its destination, and thetimes at which the packet is transmitted along its route.We define a valid schedule for each packet m as thecollection of links of a route, as well as the times oftransmissions for each of these links, so that packet m can be delivered to its destination on time. For example,consider the network shown in Fig. 1. Suppose a packetarrives at node A at time slot 1, and needs to be deliveredto node F before the end of time slot 3. One validschedule for this packet is to transmit it over link d intime slot 1, and then over link g in time slot 2. We use { ( d, , ( g, } to represent this valid schedule. Other validschedules include { ( d, , ( g, } , { ( e, , ( f, , ( g, } , etc.On the other hand, { ( d, , ( g, } is not a valid schedulebecause the packet is delivered to its destination after itsdeadline at time slot 4. The schedule { ( d, , ( g, } is notvalid because it would require node D to transmit thepacket over link g at time slot 2 before it receives thepacket at time slot 3. For each packet m , we let V ( m ) denote the set of valid schedules for m . The problem ofdeciding how to deliver packets on time then becomesone of choosing valid schedules for packets.We use X mk to denote the schedule selection for packet m . If X mk = 1 , packet m is transmitted using valid sched-ule k , and X mk = 0 , otherwise. Given the information ofall packets, the problem of maximizing the total number (cid:36) (cid:40) (cid:39) (cid:38) (cid:41)(cid:68) (cid:69)(cid:71) (cid:70)(cid:73) (cid:74)(cid:72) Fig. 1. Network topology. of successful deliveries can be formulated as the followinglinear programming problem:
Schedule:
M ax (cid:88) m,k : k ∈ V ( m ) X mk (1) s.t. (cid:88) k : k ∈ V ( m ) X mk ≤ , ∀ m ∈ M , (2) (cid:88) m,k :( l,t ) ∈ k X mk ≤ C l , ∀ l ∈ L , t ∈ { , , . . . } , (3) X mk ≥ , ∀ m ∈ M , k ∈ V ( m ) . (4)Since X mk = 1 if packet m is transmitted using validschedule k , Eq. (1) is the total number of packets that aredelivered on time. Eq. (2) states that at most one validschedule can be chosen for each packet. Eq. (3) statesthat each link can transmit at most C l packets in any timeslot. In practice, X mk can only be either 0 or 1, but ourproblem formulation allows X mk to be any real number in [0 , . Thus, the optimal solution to Schedule describes anupper bound on the total number of successful deliveries.If information of all packets is available when the sys-tem starts, the optimal solution to
Schedule can be foundby standard linear programming methods. In practice,however, packets arrive sequentially, and we need to relyon online policies that determines the values of X mk foreach arriving packet m without knowing future packetarrivals. Without the knowledge of future arrivals, it isobvious that online policies cannot always achieve theoptimal solution to Schedule . In fact, a recent work [1]has shown that, when the longest path between a sourcenode and a destination node is L , no online policy canguarantee to deliver more than L as many packets asthe optimal solution. To put this number in perspective,consider a medium-sized network with L = 8 . Even whenthe optimal solution can deliver all packets on time, thebound in the recent work states that no online policy canguarantee to deliver more than = of all packets. Such performance of online policies is unacceptable forvirtually any applications.In order to achieve good performance for online policiesin the presence of unknown future arrivals, we considerthe scenario where service providers can increase linkcapacities by, for example, upgrading network infrastruc-tures. When the link capacities are increased by R times,link l can transmit RC l packets in each time slot. Withthe increase in capacities, our problem can be rewrittenas follows: Schedule( R ): M ax (cid:88) m,k : k ∈ V ( m ) X mk (5) s.t. (cid:88) k : k ∈ V ( m ) X mk ≤ , ∀ m ∈ M , (6) (cid:88) m,k :( l,t ) ∈ k X mk ≤ RC l , ∀ l ∈ L , t ∈ { , , . . . } , (7) X mk ≥ , ∀ m ∈ M , k ∈ V ( m ) . (8)To evaluate the performance of online policies, wedefine a competitive ratio that incorporates the increasein capacities: Definition 1:
Given a sequence of packet arrivals, let Γ opt be the optimal value of (cid:80) mk : k ∈ V ( m ) X mk in Sched-ule , and Γ η ( R ) be the number of packets that are deliv-ered under an online policy η when the link capacitiesare increased by R times. The online policy η is said tobe ( R, ρ ) -competitive if Γ opt / Γ η ( R ) ≤ ρ , for any sequenceof packet arrivals.IV. A N O NLINE A LGORITHM AND I TS C OMPETITIVE R ATIO
A. Algorithm Description
In this section, we propose an online policy based onprimal-dual method and analyze the competitive ratio.We first note that the dual problem of
Schedule is:
Dual : M in (cid:88) m α m + (cid:88) l,t C l β lt , (9) s.t. α m + (cid:88) l,t :( l,t ) ∈ k β lt ≥ , ∀ m ∈ M , k ∈ V ( m ) (10) α m ≥ , ∀ m, (11) β lt ≥ , ∀ l, t, (12)where α m is the Lagrange multiplier corresponding toconstraint (2), and β lt is the Lagrange multiplier corre-sponding to constraint (3).By the Weak Duality Theorem, we have the followinglemma: Lemma 1:
Given any vectors of { α m } and { β lt } thatsatisfy the constraints (10)–(12), we have (cid:80) m α m + (cid:80) ( l,t ) C l β lt ≥ Γ opt .e now introduce our online algorithm. Our algorithmconstructs { X mk } , { α m } , { β lt } simultaneously while en-suring they satisfy all constraints in Schedule( R ) and Dual . Initially, it sets β lt ≡ . When a packet m arrives,the algorithm finds the valid schedule k ∗ that has thelargest (1 − (cid:80) l,t :( l,t ) ∈ k β lt ) among all k ∈ V ( m ) . If − (cid:80) l,t :( l,t ) ∈ k ∗ β lt ≤ , then the algorithm drops packet m and sets α m = 0 and X mk = 0 , for all k ∈ V ( m ) .On the other hand, if − (cid:80) l,t :( l,t ) ∈ k ∗ β lt > , packet m is transmitted using the valid schedule k ∗ . Our algorithmsets X mk ∗ = 1 , α m = 1 − (cid:80) l,t :( l,t ) ∈ k ∗ β lt , and updates β lt as β lt = β lt (1 + C l ) + d l − C l for all ( l, t ) ∈ k ∗ , where d l is chosen to be (1 + 1 /C l ) RC l . The complete policy isshown in Algorithm 1. Algorithm 1
Online Algorithm with Variable R Initially, α m ← , β lt ← , X mk ← . d l ← (1 + 1 /C l ) RC l , ∀ l . for each arriving packet m do k ∗ ← argmax k (1 − (cid:80) ( l,t ) ∈ k β lt ) if (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > then α m ← (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) β lt ← β lt (1 + 1 C l ) + 1( d l − C l , ( l, t ) ∈ k ∗ X mk ∗ ← . Transmit packet m using valid schedule k ∗ . else Drop packet m . end if end for B. Complexity of the Algorithm
In step 4, the algorithm finds the valid schedule k ∗ thatmaximizes (1 − (cid:80) l,t :( l,t ) ∈ k β lt ) . We now show that thisstep can be completed in polynomial time by dynamicprogramming. Before presenting the algorithm, some newnotations are given as follows. We say that packet m joinsthe network at the beginning of time slot a m , and specifiesits deadline as f m . Its source node and destination nodeare s m and d m , respectively. Therefore, a valid schedulefor m is one that can deliver a packet from node s m tonode d m between time slots a m and f m .Let Θ( n, τ ) be the smallest value of (cid:80) l,t :( l,t ) ∈ k β lt among all schedules that can deliver a packet from node s m to node n between time slots a m and τ . Θ( n, τ ) = ∞ if there is no schedule that delivers a pacekt from s m to d m between time slots a m and τ m . Step 4 of Alg. 1 isthen equivalent to finding the valid schedule that achieves (cid:80) l,t :( l,t ) ∈ k β lt = 1 − Θ( d m , f m ) . Since packet m arrivesat the beginning of time slot a m , or, equivalently, at theend of time slot a m − , we set Θ( s m , a m −
1) = 0 and Θ( n, a m −
1) = ∞ for ∀ n (cid:54) = s m .There are only two different ways to deliver a packetto node n by the end of time slot τ : The first is to deliver the packet to n by time slot τ − , in which case (cid:80) l,t :( l,t ) ∈ k β lt = Θ( n, τ − . The second is to deliver thepacket to one of n ’s neighbors, say, node q , by time slot τ − , and then forward the packet along the link l qn from q to n at time slot τ . In this case, (cid:80) l,t :( l,t ) ∈ k β lt =Θ( q, τ −
1) + β l qn τ . Therefore, we have Θ( n, τ ) = min (cid:26) Θ( n, τ − , Θ( q, τ −
1) + β l qn τ , q is a neighbor of n .Based on the above recursive equation, we design analgorithm for computing Θ( n, τ ) . The detailed algorithmis shown in Algorithm 2, where we also use Sch ( n, τ ) todenote the schedule that achieves Θ( n, τ ) . Algorithm 2
Dynamic Programming for each arriving packet m do Θ( s m , a m − ← Θ( n, a m − ← ∞ , ∀ n (cid:54) = s m Sch ( n, a m − ← φ, ∀ n for τ = a m to f m do for node n do Θ( n, τ ) ← Θ( n, τ − Sch ( n, τ ) ← Sch ( n, τ − for node n ’s neighbor q do if Θ( q, τ −
1) + β l qn τ < Θ( n, τ ) then Θ( n, τ ) ← Θ( q, τ −
1) + β l qn τ } Sch ( n, τ ) ← Sch ( q, τ − ∪ { ( l qn , τ ) } end if end for end for end for end for In Alg. 2, the inequality Θ( q, τ −
1) + β l qn τ < Θ( n, τ ) isonly evaluated once for any link and time slot. Let E bethe number of links in the system. Suppose the number oflinks is larger than the number of nodes, and f m − a m +1 ≤ T , for all m , then the complexity of Alg. 2 is O ( ET ) . C. Competitive Ratio Analysis
Before analyzing the performance of Algorithm 1, wefirst establish a basic property of the values of β lt . Lemma 2:
Let β lt [ n ] be the value of β lt after n packetsare scheduled to use link l at time t . Then, β lt [ n ] = ( 1 d l − d n/RC j l − . (13) Proof:
First, note that the value of β lt is only changedwhen Algorithm 1 uses link l at time t to transmit apacket. Therefore, the value of β lt only depends on thenumber of packets that are scheduled to use link l at time t . We then prove (13) by induction. Initially, when n = 0 , β lt [0] = 0 = ( d l − )( d l − and (13) holds.Suppose (13) holds for the first n packets. When the ( n + 1) -th packet is scheduled for link l at time t , we have lt [ n + 1] = β lt [ n ](1 + 1 C l ) + 1( d l − C l = 1( d l − d n/RC l l − C l ) + 1( d l − C l = 1 d l − d n/RC l l (1 + 1 C l ) − We select d l = (1 + C l ) RC l , and therefore β lt [ n + 1] = 1( d l −
1) [ d ( n +1) /RC l l − , and (13) still holds for n + 1 . Thus, by induction, (13)holds for all n .We now establish the competitive ratio of Algorithm 1. Theorem 1:
Let C min := min C l , d min := (1 +1 /C min ) RC min , and L be the longest path between asource node and a destination node, that is, all validschedules have | k | ≤ L , for all m ∈ M , k ∈ V ( m ) . Algo-rithm 1 produces solutions that satisfy all constraints in Schedule( R ) and Dual . Moreover, Algorithm 1 is ( R, Ld min − ) -competitive, which converges to ( R, Le R − ) -competitive, as C min → ∞ . Proof:
First, we show that the dual solutions { α m } and { β lt } satisfy constraints (10) to (12). Initially, wehave β lt = 0 . By Lemma 2, β lt ≥ holds. Since step 6is only used when (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > , α m ≥ holds.From step 4 and 6, we know that α m + (cid:80) ( l,t ) ∈ k β lt ≥ (1 − (cid:80) ( l,t ) ∈ k β lt ) + (cid:80) ( l,t ) ∈ k β lt = 1 . Thus (10) to (12)hold.Next, we show { X mk } satisfies constraints (6) to (8).By step 4, the algorithm picks at most one schedule k ∗ for packet m , constraint (6) holds. With Lemma 2, β lt =1 when RC l packets use link l at time t . Since a validschedule including ( l, t ) will be chosen for packet m onlywhen (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > , all ( l, t ) in the chosen validschedule must have β lt < , and therefore the number ofpackets transmitted over link l at time t must be less than RC l . Thus, at any time t , there are at most RC l packetsusing link l . Constraint (7) holds. By initialization andstep (8), constraint (8) holds.We derive the ratio between (cid:80) m α m + (cid:80) ( l,t ) C l β lt and (cid:80) mk X mk . Initially, both are equal to 0. We consider theincreasing amount for both when a new packet m arrivesat the network. We use ∆ P ( R ) to denote the change of (cid:80) mk X mk , and ∆ D to denote the change of (cid:80) m α m + (cid:80) ( l,t ) C l β lt .If packet m is dropped, both ∆ P ( R ) and ∆ D are 0. Ifpacket m is accepted and transmitted using valid schedule k ∗ , we have X mk ∗ = 1 . Thus, ∆ P ( R ) = 1 . On the otherhand, ∆ D is increased as: ∆ D = α m + (cid:88) ( l,t ) ∈ k ∗ C l ∆ β lt =(1 − (cid:88) ( l,t ) ∈ k ∗ β lt ) + (cid:88) ( l,t ) ∈ k ∗ ( β lt + 1( d l − C l )=1 + (cid:88) ( l,t ) ∈ k ∗ d l − ≤ Ld min − Therefore, for each packet arrival, the ratio be-tween ∆ D and ∆ P ( R ) is no larger than Ld min − if ∆ D > . When the algorithm terminates, we have (cid:80) m α m + (cid:80) ( l,t ) C l β lt (cid:80) mk X mk ≤ Ld min − . By Lemma 1, Γ opt (cid:80) mk X mk ≤ Ld min − , and the competitive ratio of Algorithm 1is ( R, Ld min − ) . When C min → ∞ , d min = (1 + C min ) RC min → e R , and the competitive ratio of Algo-rithm 1 converges to ( R, Le R − ) .There are several important implications of Theorem 1.First, without increasing capacity, that is, when R = 1 , thecompetitive ratio of our policy is (1 , O ( L )) . In compar-ison, the online algorithm proposed in the recent work[1] focuses on the special case of R = 1 and has acompetitive ratio of (1 , O ( L log L )) . Therefore, our algo-rithm is asymptotically better than the online algorithmin [1]. Second, this theorem allows us to quantify theamount of capacity needed to a certain performanceguarantee. Suppose the optimal solution to Schedule indeed delivers all packets. In order to guarantee that − θ of the packets are transmitted to their destinationsbefore their deadlines, Theorem 1 states that we onlyneed to increase all link capacities by R θ times so that Le Rθ − ≤ / (1 − θ ) = 1 + θ − . Therefore, we have R θ = ln ( L ( θ −
1) + 1) ≤ ln L + ln θ . For example, ifwe are required to deliver of the packets and thelongest path consists of 10 hops, then we need to increasecapacity by 6.9 times.V. A T HEORETICAL L OWER B OUND FOR C OMPETITIVE R ATIO
In Section IV, we showed that our policy is ( R, Le R − ) -competitive. In this section, we will establish alower bound for the competitive ratio of online policies. Theorem 2:
Any online algorithm cannot be better than ( R, L − e R ( L +1) e R − L ) -competitive. Proof:
We design a network as shown in Fig 2. Westart to construct the network from an up-link tree, whichis shown as the white nodes in Fig 2. Root is markedas node D and it is the destination of all packets. Thereare N levels of non-root nodes with N nodes in eachlevel. Each node is connected to one node in the nextlevel. Nodes do not share parent except the N -th levelnodes share the same root node. At the j -th level, where ≤ j ≤ N , there are (cid:0) NN +1 − j (cid:1) extra nodes, which is shown (cid:21) (cid:39)(cid:49)(cid:22) (cid:22127) (cid:22127) (cid:22127) (cid:22127) (cid:22127) (cid:79) (cid:20)(cid:20) (cid:79) (cid:20)(cid:21) (cid:79) (cid:20)(cid:22) (cid:79) (cid:20)(cid:23) (cid:79) (cid:20)(cid:49) (cid:79) (cid:21)(cid:20) (cid:79) (cid:21)(cid:21) (cid:79) (cid:21)(cid:11)(cid:49)(cid:16)(cid:20)(cid:12) (cid:79) (cid:49)(cid:20) (cid:79) (cid:22)(cid:11)(cid:49)(cid:16)(cid:21)(cid:12) (cid:79) (cid:22)(cid:20) (cid:79) (cid:22)(cid:21) (cid:39) (cid:22127) (cid:22127) (cid:22127) (cid:22127) (cid:22127)(cid:22127)(cid:22127) (cid:22127) (cid:20) (cid:86)(cid:87) (cid:3)(cid:79)(cid:72)(cid:89)(cid:72)(cid:79) (cid:21) (cid:81)(cid:71) (cid:3)(cid:79)(cid:72)(cid:89)(cid:72)(cid:79) (cid:22) (cid:85)(cid:71) (cid:3)(cid:79)(cid:72)(cid:89)(cid:72)(cid:79) (cid:11)(cid:49)(cid:16)(cid:20)(cid:12) (cid:87)(cid:75) (cid:49) (cid:87)(cid:75) Fig. 2. Network topology for lower bound analysis (cid:20) (cid:21) (cid:39)(cid:49)(cid:22) (cid:22127) (cid:22127) (cid:22127) (cid:22127) (cid:22127) (cid:79) (cid:20)(cid:20) (cid:79) (cid:20)(cid:21) (cid:79) (cid:20)(cid:22) (cid:79) (cid:20)(cid:23) (cid:79) (cid:20)(cid:49) (cid:79) (cid:21)(cid:20) (cid:79) (cid:21)(cid:21) (cid:79) (cid:21)(cid:11)(cid:49)(cid:16)(cid:20)(cid:12) (cid:79) (cid:49)(cid:20) (cid:79) (cid:22)(cid:11)(cid:49)(cid:16)(cid:21)(cid:12) (cid:79) (cid:22)(cid:20) (cid:79) (cid:22)(cid:21)
Fig. 3. Simplified network topology for lower bound analysis as the black nodes in Fig 2, with each node connectingto an unique set of N + 1 − j nodes in this level. Forexample, there is one black node connected to all whitenodes in level 1, and there are N black nodes connectedto white nodes in level 2, where each of these black nodesis connected all but one white nodes in level 2. Likewise,there are (cid:0) NN − (cid:1) black nodes connected to white nodes inlevel 3, with each black node connected to N − whitenodes in level 3, and no two black nodes are connectedto the same subset of white nodes.Next, we describe packet arrivals. Packets only arriveat black nodes. Of all black nodes connected to the samelevel of white nodes, only one black node has packetarrival. Let W j be the set of white nodes in j -th levelwhich connects to the black node with packet arrivals.The black nodes with packet arrivals are chosen such thatall nodes in W j +1 are connected to those in W j . Fig 3 isa simplified network of Fig 2, where we omit the blacknodes with no packet arrival and marked each black nodewith a number from 1 to N .Packets arrive at nodes , , ..., N . Their destination is node D . Each link in the network has capacity C . At thebeginning of time slot , there are C packets arriving atnode . Node is connected to N links: l , l , · · · , l N . Atthe beginning of time slot , there are C packets arrivingat node . Node is connected to N − links: l , l , · · · , l N − . Similarly for nodes , , · · · . At the beginningof time N , there are C packets arriving at node N . Thedeadline of all packets is N +1 . Node N is connected onlyto link l N .When one knows which black nodes have packet ar-rivals, the offline optimal algorithm is to transmit thefirst C packets through link l and the following links,the second C packets through link l and the followinglinks, . . . , and the N -th C packets through link l N andthe following link. The total number of delivered packetsis N C .Next we consider the online algorithm when all links’capacity is increased by R times. Since online policies donot know which black nodes will have packet arrivals,the optimal online policy is to distribute packets evenlyamong all connected links. That is, at time , each oflinks l i , i = 1 , , · · · , N , transmit C/N packets. At time ,each of link l i , i = 1 , , · · · , ( N − , transmits C/ ( N − packets. At time K , link l Ki , i = 1 , , · · · , ( N − K + 1) ,transmits C/ ( N − K + 1) packets. For simplicity, we callthe routes from node to node D through l i route r i .If all packets arrive at node K are accepted, routes r i , i = K, K + 1 , · · · , N have the same load on each link.When any link on a single route reaches its capacity, theroute cannot be used for future arrival packets. Supposethe route gets over-loaded at time K + 1 , that is, packetsarrive at node K are accepted and packets arrive at node K + 1 are not fully accepted. The maximum load of asingle link on route r N is at most CN + CN − + · · · + CN − K +1 and at least CN + CN − + · · · + CN − K . We then have: C ( 1 N + 1 N − N − . . . + 1 N − K + 1 ) ≤ RC, and C ( 1 N + 1 N − N − . . . + 1 N − K ) ≥ RC.
Since (cid:90) N +1 N − K +1 x dx < ( 1 N + 1 N − N − . . . + 1 N − K + 1 ) , and (cid:90) NN − K − x dx > ( 1 N + 1 N − N − . . . + 1 N − K ) . We have: log( N + 1) − log( N − K + 1) = log N + 1 N − K + 1 < R, and log( N ) − log( N − K −
1) = log NN − K − > R. hen we can derive the value of K as: N − Ne R − ≤ K ≤ N + 1 − N +1 e R . The total number of accepted packetsis in the range (( N − Ne R − C, ( N + 2 − N +1 e R ) C ) .Thus the competitive ratio of an online policy is at best ( R, NN +2 − N +1 eR ) . In Fig. 2, the longest path in the networkis between the leftmost black node and the sink, whichhas length L = N + 1 . The competitive ratio can then berewritten as ( R, L − e R ( L +1) e R − L ) .Let us once again consider the scenario where onlinepolicies need to guarantee to deliver at least − θ asmany packets as the optimal solution. Theorem 2 statesthat any online policy needs to increase its link capacitiesby at least R θ times so that L − e Rθ ( L +1) e Rθ − L ≤ θ − .Solving this equation, and we have R θ needs to be atleast ln L + ln θ − ln( L + 2 θ − . In comparison, our policyonly needs to increase link capacities by (ln L +ln θ ) timesto ensure the delivery of − θ as many packets as theoptimal solution. Therefore, the capacity requirement ofour policy is at most ln( L + 2 θ − away from the lowerbound. Suppose we fix the ratio between L and θ , and letthem both go to infinity, then we have (ln L +ln θ ) / (ln L +ln θ − ln( L + 2 θ − → . Therefore, when both L and θ are large, our policy at most requires twice as muchcapacity as the theoretical lower bound.VI. A N O RDER -O PTIMAL O NLINE P OLICY WITH F IXED R = 1 We have shown that Alg. 1 is ( R, Ld min − ) -competitive. Without increasing link capacity, i.e, R = 1 ,the algorithm is (1 , Le − ) -competitive, as C min → ∞ .While the competitive ratio of Alg. 1 is an order betterthan that of the online policy in the previous work [1], itstill fails to achieve the theoretical bound of (1 , O (log L )) -competitive. In this section, we propose another onlinealgorithm and prove that it achieves the theoretical boundwhen R = 1 . A. Algorithm Description
Similar to the design of Alg. 1, we aim to designan algorithm that constructs { X mk } , { α m } , { β lt } whileensuring they satisfy all constraints in Schedule and
Dual .The algorithm is described in Alg. 3. One can see that Alg.3 is very similar to Alg. 1, and their only difference lie inthe update rules for β lt . Specifically, let β lt [ n ] be the valueof β lt when link l serves a total number of n packets attime t . Then Alg. 3 chooses the value of β lt [ n ] as: β lt [ n ] = L ( e L +1 − e nCl − , if n ≤ C l ln L +1 ; e ( nCl − L +1) , if n ≥ C l ln L +1 . (14)To illustrate the difference in β lt , we plot the values of β lt [ n ] for a link with C l = 1000 under the two policies inFig. 4, where we consider the two cases L = 8 and L = 64 for Alg. 3. As can be shown in the figure, when n is small,Alg. 3 increases the value of β lt much slower than Alg. β Variable RFixed R with L=8Fixed R with L=64
Fig. 4. Values of β lt under different policies. β lt slower when L islarger. Recall that both Alg. 1 and Alg. 3 only schedule apacket when max k (1 − (cid:80) ( l,t ) ∈ k β lt ) > , or, equivalently, min k (cid:80) ( l,t ) ∈ k β lt < . By increasing β lt slower when n issmall, Alg. 3 ensures that more packets with long routescan be accepted, especially when the network is lightlyloaded. Algorithm 3
Online Algorithm with Fixed R = 1 Initially, α m ← , β lt ← , X mk ← . for each arriving packet m do k ∗ ← argmax k (1 − (cid:80) ( l,t ) ∈ k β lt ) if (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > then α m ← (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) for each ( l, t ) ∈ k ∗ do if total number of packets n at time t on link l : n ≤ C l ln L +1 then β lt ← L ( e L +1 − e nCl − , else β lt ← e ( nCl − L +1) end if end for X mk ∗ ← . Transmit packet m using valid schedule k ∗ . else Drop packet m . end if end for B. Competitive Ratio Analysis
We now prove that Alg. 3 achieves the theoreticalbound in [1] by being (1 , O (log L )) -competitive. Lemma 3:
Let C min := min C l . In Algorithm 3, eachtime a new packet is scheduled, the ratio between thehange of Schedule and
Dual is bounded by L + 1) + BC min , where the value of B is independent of C min . Proof:
If a new packet is admitted to the network, theincreasing amount of
Dual is ∆ D = α m + (cid:88) ( l,t ) ∈ k ∗ C l ∆ β lt =1 + (cid:88) ( l,t ) ∈ k ∗ ( C l ∆ β lt − β lt ) We define β ( x ) as β ( x ) = L ( e L +1 − e x − , if x ≤ L +1 ; e ( x − L +1) , if x ≥ L +1 . (15)Note that β lt [ n ] = β ( nC l ) . By using Taylor Sequence, wethen have ∆ β lt [ n ] := β lt [ n + 1] − β lt [ n ] = β ( n + 1 C l ) − β ( nC l ) ≤ C l β (cid:48) ( nC l ) + (cid:15) C l β (cid:48)(cid:48) ( nC l ) , for some bounded constant (cid:15) < ∞ , where β (cid:48) and β (cid:48)(cid:48) arethe first and second derivative of β , respectively. We notethat the function β ( x ) is continuous for all x , and infinitelydifferentiable for all x except at the point x := L +1 .At the point x , we define β (cid:48) ( x ) = lim x → x +0 β (cid:48) ( x ) and (cid:15)β (cid:48)(cid:48) ( x ) = lim x → x +0 (cid:15)β (cid:48)(cid:48) ( x ) . This ensures that the aboveinequality still holds.By (14) we know that n ≤ C l ln L +1 if and only if β lt [ n ] ≤ L .If x = nC l ≤ L +1 , then β (cid:48) ( x ) = β (cid:48)(cid:48) ( x ) = e x L ( e L +1 − .We have: C l ∆ β lt [ n ] − β lt [ n ] ≤ C l ( C l e nCl ) + (cid:15) ( C l ) e nCl ) − ( e nCl − L ( e L +1 − ≤ (cid:15) C l e nCl L (1 + L +1 − ≤ ln L + 1 L (1 + (cid:15) C l e ) Let B = (cid:15)e ln L +1 L , then C l ∆ β lt [ n ] − β lt [ n ] ≤ ln L + 1 L + B C min , (16)when nC l ≤ L +1 . On the other hand, If x = nC l ≥ L +1 , then β (cid:48) ( x ) =(ln L + 1) β ( x ) and β (cid:48)(cid:48) ( x ) = (ln L + 1) β ( x ) . We have: C l ∆ β lt [ n ] − β lt [ n ] ≤ C l [ ln L + 1 C l β lt [ n ] + (cid:15) ( ln L + 1 C l ) β lt [ n ]] − β lt [ n ] ≤ ln L · β lt [ n ] + 1 C l (cid:15) (ln L + 1) β lt [ n ] Let B = (cid:15) (ln L + 1) , then C l ∆ β lt [ n ] − β lt [ n ] ≤ (ln L + B C min ) β lt [ n ] , (17)when nC l ≥ L +1 .If packet m is transmitted using valid schedule k ∗ , wehave X mk ∗ = 1 . Thus, ∆ P = 1 . On the other hand, ∆ D is increased as: ∆ D =1 + (cid:88) ( l,t ):( l,t ) ∈ k ∗ C l ∆ β lt − β lt ≤ (cid:88) ( l,t ):( l,t ) ∈ k ∗ ,β lt ≤ L C l ∆ β lt − β lt + (cid:88) ( l,t ):( l,t ) ∈ k ∗ ,β lt ≥ L C l ∆ β lt − β lt From (16) and (17) we have: ∆ D ≤ (cid:88) ( l,t ):( l,t ) ∈ k ∗ ,β lt ≤ L ( ln L + 1 L + B C min )+ (cid:88) ( l,t ):( l,t ) ∈ k ∗ ,β lt ≥ L ((ln L + B C min ) β lt ) From Algorithm 3 step 4 we know that (cid:80) β lt ≤ , thuswe have ∆ D ≤ L + 1 + B LC min ) + (ln L + B C min )=2 + 2 ln L + B + B C min , and the proof is complete. Theorem 3:
Algorithm 3 produces solutions that satisfyall constraints in
Schedule and
Dual . Moreover, it is (1 , L )) -competitive, as C min → ∞ . Proof:
First, we show that the dual solutions { α m } and { β lt } satisfy constraints (10) to (12). Initially, wehave β lt = 0 . By (14), β lt ≥ holds. Since step 5 isonly used when (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > , α m ≥ holds.From step 3 and 5, we know that α m + (cid:80) ( l,t ) ∈ k β lt ≥ (1 − (cid:80) ( l,t ) ∈ k β lt ) + (cid:80) ( l,t ) ∈ k β lt = 1 . Thus (10) to (12)hold.Next, we show { X mk } satisfies constraints (2) to (4).By step 3, the algorithm picks at most one schedule k ∗ for packet m , constraint (2) holds. With (14), when thenumber of packets on link l at t is C l , we have β lt = 1 .Also, since a packet is scheduled if (1 − (cid:80) ( l,t ) ∈ k ∗ β lt ) > ,we have β lt < for all ( l, t ) ∈ k ∗ . Therefore, the numberf packets transmitted on link l at any time t is at most C l . Constraint (3) holds. By initialization and step (13),constraint (4) holds.When a new packet m arrives, it will either be droppedor scheduled. If it is dropped, both ∆ P and ∆ D are 0. If itis scheduled, both (9) and (1) increase. With Lemma 3,theratio between ∆ P and ∆ D is bounded by L ) + BC min . Therefore the competitive ratio of Algorithm 3 is (1 , L ) + BC min ) → (1 , L )) , as C min → ∞ .Thus, comparing with the result in [1], Algorithm 3achieves the optimal competitive ratio when R = 1 .VII. A F ULLY D ISTRIBUTED P ROTOCOL FOR I MPLEMENTATION
The two algorithms that we have proposed so far areboth centralized algorithms. Specifically, when a packetarrives at a node, the node needs to have completeknowledge of all β lt of all links to find a valid schedule.Such information is usually infeasible to obtain. In thissection, we propose a distributed protocol based on thedesign of Algorithm 1.In our distributed protocol, the task of transmitting apacket to its destination is decomposed into two parts:First, when a packet arrives at a node, the node deter-mines a suggested schedule based on statistics of pastsystem history. This suggested schedule consists of theroute for forwarding the packet, as well as a local deadlinefor each link. After determining the suggested schedule,the node simply forwards it to the first link of the route.On the other hand, when a link receives a packet alongwith a suggested schedule, the link tries to forward thepacket to the next link in the suggested schedule before itslocal deadline. The link drops the packet when it cannotforward the packet on time.To facilitate this protocol, each link keeps track of itsown β lt , which reflects the number of packets that arescheduled to be transmitted over link l at time t . Thevalue of β lt changes over time, as link l schedules moreand more packets to be transmitted at time t . Therefore,we define β lt, ˆ t as the value of β lt when the current time is ˆ t . Each link then measures γ l,τ as the average of β lt,t − τ .In other words, when the current time is t , the expectedvalue of β lt is γ l,t − t . Link l broadcasts its γ l,τ periodicallyso that all nodes can estimate the values of β lt .We now describe how a node determines a sug-gested schedule upon the arrival of a packet. Supposea packet arrives at time t . Following the Alg. 1, thenode would like to find a valid schedule that maximizes (1 − (cid:80) l,t :( l,t ) ∈ k β lt ) . In practice, the node does not knowthe exact value of β lt . However, it knows that the expectedvalue of β lt is γ l,t − t . In our protocol, the node assumesthat β lt = γ l,t − t , and then finds a valid schedule k ∗ that maximizes (1 − (cid:80) ( l,t ) ∈ k γ l,t − t ) . Similar to Alg. 1,the node drops the packet if (1 − (cid:80) ( l,t ) ∈ k ∗ γ l,t − t ) ≤ . If (1 − (cid:80) ( l,t ) ∈ k ∗ γ l,t − t ) > , then the node puts information of k ∗ into the header of the packet, and forwards thepacket to the first link in k ∗ . Algorithm 4
Distributed Implementation: Schedule Sug-gestion for Each Node for each arriving packet m do t ← current time k ∗ ← argmax k (1 − (cid:80) ( l,t ) ∈ k γ l,t − t ) if (1 − (cid:80) ( l,t ) ∈ k γ l,t − t ) > then Put information of the suggested schedule k ∗ inthe header of packet m . Forward the packet to the first link in k ∗ . else Drop packet m . end if end for Since the actual value of β lt can be different from γ l,t − t , there is no guarantee that a packet can be de-livered on time using the valid schedule k ∗ even if (1 − (cid:80) ( l,t ) ∈ k ∗ γ l,t − t ) > . Therefore, when a node determinesa valid schedule k ∗ for a packet, the valid schedule k ∗ istreated only as a suggestion for links in k ∗ . Specifically,if k ∗ contains an entry ( l ∗ , t ∗ ) , then the link l ∗ interprets k ∗ as a requirement that l ∗ needs to forward the packetto the next link before t ∗ , or drops the packet. When l ∗ obtains the packet, it still has the freedom to choose whento forward the packet, as long as the packet is forwardedbefore time t ∗ .Next, we discuss how each link determines the actualtime to transmit each packet. Obviously, each link l ∗ knows its own β l ∗ t . From the design of Alg. 1, we can seethat Alg. 1 prefers to transmit packets when β lt is small.Our proposed policy is based on this principle. When alink l ∗ receives a packet, it finds the entry ( l ∗ , t ∗ ) from thevalid schedule k ∗ specified in the header of the packet.Link l ∗ then finds a time t tx between the current timeand t ∗ that has the smallest β l ∗ t , and transmits the packetat time t tx . Alg. 5 describes the details of the policy forpacket transmission.VIII. S IMULATION
In this section, we evaluate the performance of ourpolicies by simulation. We compare our algorithms withEDF policy and the policy, which we call Mao-Koksal-Shroff (MKS) online algorithm, proposed in [1]. BothEDF policy and MKS online algorithm focus on packetscheduling, and are applicable only when the route ofthe packet is given. For these two policies, we assumethat each packet is routed through the shortest path.We first consider a small network as shown in Fig 5.The network has 9 nodes from node 1 to node 9. Thereare directed arrows showing the directed links betweennodes. All links have the same capacity C = 1 . Weassume that there are 1000 packets arriving at the system.For each packet, the source node is chosen uniformly at lgorithm 5 Distributed Implementation: Packet Trans-mission for Each Link for each packet m do Upon m ’s arrival at a link l ∗ , the link reads scheduleinformation k ∗ from the header of the packet. Let t ∗ be the local deadline such that ( l ∗ , t ∗ ) ∈ k ∗ . t ∗ tx ← argmin t tx : t tx ≤ t ∗ β l ∗ ,t tx if β l ∗ ,t ∗ tx < then β l ∗ t ∗ tx ← β l ∗ ,t ∗ tx (1 + 1 C l ∗ ) + 1( d l ∗ − C l ∗ Transmit packet m on link l ∗ at time t ∗ tx . else Drop packet m . end if end for (cid:24) (cid:25)(cid:27) (cid:28)(cid:21) (cid:22)(cid:23)(cid:26)(cid:20) Fig. 5. Network topology for a small network random between node 1 to node 6, and the destinationis chosen uniformly at random between node 7 to node9. The inter-arrival time between packets are chosen tobe 0 with probability 0.7 and 1 with probability 0.3. Thedeadline of each packet equals its arrival time plus a slacktime. The slack time is chosen uniformly from integersbetween 2 and 6.Simulation results for different values of R are shownin Fig. 6. From the result, we can see that all our threepolicies outperform two other current policies. From thefigure, we can see that all our policies are able to deliverall packets when R is 2. On the other hand, EDF is ableto deliver all packets when R = 3 , and MKS can deliverall packets only when R is as large as 6. We also notethat both EDF and MKS are centralized policies. The factthat our distributed algorithm performs better than thesetwo centralized policies further highlights the superiorityof our algorithms.Next, we consider that different links can have differentcapacities. Since MKS requires all links to have the samecapacity, we only compare our policies against EDF. Thenetwork topology is also shown in Fig 5. We assume that, D e li v e r r a t i o Online Algorithm with Variable REDF AlgorithmOnline Algorithm with Fix R=1MKS AlgorithmDistributed Algorithm
Fig. 6. Deliver ratio comparison when all links have the same capacity. D e li v e r r a t i o Online Algorithm with Variable ROnline Algorithm with Fix R=1EDF AlgorithmDistributed Algorithm
Fig. 7. Deliver ratio comparison when different links have differentcapacities. when R = 1 , the link capacity are integers uniformlychosen from 5 to 10. There are 10000 i.i.d packets tobe delivered. At the beginning of each time slot, thereare a certain number of packets arriving the system. Thesource node and destination node are both chosen fromnode 1 to node 9 with equal probability and destinationnode is not allowed to be the same with source node.The number of packets is randomly chosen between 100and 500. Each packet has a slack time between arrival anddeadline, which is uniformly chosen from [2 , . The resultis shown in Fig 7. Once again, we see that our policies,including the distributed algorithm, perform much betterthan EDF in most cases.X. C ONCLUSION
In this paper, we study the multi-hop network schedul-ing problem with end-to-end deadline and hard transmis-sion rate requirement. Given the capacity of each link inthe network, we aim to find out how much capacity weneed to increase to guarantee the required ratio of packetscan be successfully transmitted to its destination before itsdeadline without knowing the packet arrival sequences inadvance.We have proposed an online algorithm which works forboth fix route and non-fix route network. The algorithmis proved to be ( R, Le R − ) -competitive, where L is thelength of the longest path. We have also showed thatthe complexity of our algorithm is O ( ET ) , where E isthe total number of links and T is the largest slack time.Next, we have showed that any online algorithm cannotbe better than ( R, L − e R ( L +1) e R − L ) -competitive. When both L and required deliver rate are large, our policy requiresat most twice as much capacity as the lower bound.In addition, We have proposed an online algorithm forfixed capacity network. When the capacity cannot beincreased, our algorithm is proved to be (1 , O (log L )) -competitive, which is also an order-optimal policy. Forpractical implementation of our centralized algorithm,we have proposed a heristic for distributed algorithmso that each node can make decisions without requiringreal-time information from all other nodes. In additionto the theoretical results, we compare our policies withtwo other online policies, including the widely-used EDFpolicy and a recent proposed policy, by simulation. Theresults show that the performance of our policies arebetter than the other two policies. Also the result showsthat the distributed algorithm still provide a good deliveryratio. X. A CKNOWLEDGMENT
This material is based upon work supported in part bythe U. S. Army Research Laboratory and the U. S. ArmyResearch Office under contract/grant number W911NF-15-1-0279 and NPRP Grant 8-1531-2-651 of Qatar Na-tional Research Fund (a member of Qatar Foundation).R
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